Two hard rubber spheres, each of mass m=15.0 g, are r\ubbed...

Key Concepts

Coulomb's Law

Coulomb's law describes the electrostatic interaction between charged particles. It states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This fundamental principle is used to determine the magnitude of repulsive or attractive forces acting between the charged spheres.

Equilibrium of Forces

In systems at equilibrium, the net force acting on each object is zero. This concept implies that all forces, such as gravitational, tension, and electrostatic forces, must balance each other out. Analyzing the equilibrium conditions ensures that the sum of the vertical and horizontal force components are both zero, which is critical in determining unknown quantities like charge or tension.

Force Decomposition into Components

Forces acting on objects in equilibrium can be resolved into perpendicular components, typically vertical and horizontal. This technique simplifies the analysis of force balances by allowing the separate equilibrium conditions to be applied to each axis. It is a key tool for solving problems where forces are applied at angles, such as tension in strings at an angle due to electrostatic repulsion.

Geometric Relationships in Equilibrium Systems

Geometrical analysis involves using the configuration of the system, such as angles and distances, to relate the forces and displacements. In equilibrium problems, geometry helps in determining the distances between forces, the angles of attraction or repulsion, and the relationship between those angles and the magnitudes of the forces acting on the system.

Charge Distribution Ratios

Understanding the relationship between differing amounts of charge, such as one charge being a multiple of another, is crucial when applying Coulomb's law to systems with non-uniform charge distributions. This concept allows for the establishment of proportional relationships that can simplify the process of finding individual charge values based on the known ratio.

Transcript

00:01 So according to this exercise, there are two rubber spheres that we rub with fur in order to induce static charge on them.

00:11 These two spheres, they have mass 15 grams or 0 .015 kilograms.

00:20 They are suspended by strings of length 5 centimeters or 0 .05 meters.

00:26 And the strings they have the other distance of 3 centimeters or 0 .3 meters and after rubbing the two spheres they acquire charge one acquires charge twice the other and at equilibrium they are suspended at an angle of theta equals 10 degrees with the vertical okay and the the exercise asks us to find out the charge on each of the two spheres.

01:02 Okay, the two formulas that we need, typically, for the electric force, the electrostatic force on a charge, it's just that all we need to do is multiply the charge of the particle of the body with the electric field at the location of the body.

01:23 Okay.

01:25 Then we also need the the formula for the vector electric field generated due to a point charge.

01:33 This is the typical one of four pi epsilon a times q over r squared.

01:38 R hat, r again, is the relative distance, meaning the vector distance from the charge to the observation point at the point that we measure or calculate the electric field.

01:53 Again, it starts from the charge and it goes to the observation.

01:56 Point it points towards the observation point not towards the charge the other way okay and q is literally here to indicate that this is a different charge than this one q literally is the one that generates the electric field q is capital is the one that feels that at which the electric fault is exerted to okay if we combine the two we get the typical formula for the electric for the for the for the for the static force felt by a point charge due to electric fuel generated by again by a point charge and we see here two times q.

02:39 Okay, so what do we have here? we have two point charges accepting electrostatic force on each other and quite naturally what we notice is that the magnitude of the electrostatic force basically if we if we replace this q litter and q capital with um with let's say q, let's say q for the one on the left and two eyes two times q for the one on the right so yeah if we place inside there we see that the magnitude of the or the force felt by both the charges is exactly the same but it has opposite direction this is actually what one might have expected if we thought of the problem in terms of the newton's third law action and reaction opposite and equal reaction felt by a by a two but is that they the feel force and the exert force simultaneously anyway okay we do have now one expression for the electric electric static force felt by any of the two any of the two point charges it is quite important here that this is so it doesn't even matter whether q is negative or positive.

04:02 We know immediately that the electrostatic force here points to the left and here to the right, it points to the right, plus x minus x.

04:13 Because again this is squared, this is a positive quantity.

04:17 This is not the case, of course, for the electric field.

04:22 The electric field depends on the type of the charge.

04:25 If q was negative, for example, the electric field here, we have been pointing there but again q is negative and for the force will be pointing to the left again in any case we have now an expression for the electric electric static force acceded on both the objects and it has exactly the same the same magnitude okay now let's decompose the forces let's draw a diagram decomposing the forces exerted on any of the two charges let's take the left one we have the force of gravity minus mass times the acceleration of gravity and y hat because we take that the the positive y axis is pointing up therefore since the weight points down it's negative okay and we have the electrostatic force pointing to the left and we have another another force from the string pointing diagonally up and to the right okay we have equilibrium.

05:35 Equilibrium.

05:39 Therefore, the sum of forces is zero.

05:42 Sum of forces is zero.

05:46 The vector is zero...

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